On some iterated means arising in homogenization theory. (English) Zbl 1099.26002

The authors consider iterations of arithmetic and power means (in particular the arithmetic, geometric and harmonic means) and discuss methods for determining their limits. In particular, they study the iterated means \((A_mP_m)^ m\) and \((P_mA_m)^ m\), in which \(A_m(x)=\alpha x+\beta\) and \(P_m(x)=(\gamma x^ {1/1-p}+\delta )^ {1-p}\), where \(\gamma =v^ {1/mn}\), \(\alpha =v^{ n-1/mn}\) and \(\gamma +\delta =\alpha +\beta =1\). It turns out that these iterated means converge to the same limit, which is in general difficult to find, but in the case \(p=2\) it is known to be equal to \(1+nv(x-1)/(n+(1-v)(x-1))\). This result was proved in the PhD thesis of the first author by obtaining explicit formulae for the iterated means. In the paper under review an alternative (matrix) proof of these formulae is given. Moreover, the case \(n=1\) is treated in detail, and some results are established for the general case and the limiting cases.


26A18 Iteration of real functions in one variable
35M20 PDE of composite type (MSC2000)
26D99 Inequalities in real analysis


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